The boundary layer equations for plane, incompressible, and steady flow are 

$$\matrix{ {u{{\partial u} \over {\partial x}} + v{{\partial u} \over {\partial y}} = - {1 \over \varrho }{{\partial p} \over {\partial x}} + v{{{\partial ^2}u} \over {\partial {y^2}}},} \cr {0 = {{\partial p} \over {\partial y}},} \cr {{{\partial u} \over {\partial x}} + {{\partial v} \over {\partial y}} = 0.} \cr }$$

Boundary Layer Theory

Recent advances in wet adhesives: Adhesion mechanism, design principle and applications

On-skin electrodes function as an ideal platform for collecting high-quality electrophysiological (EP) signals due to their unique characteristics, such as stretchability, conformal interfaces with skin, biocompatibility, and wearable comfort. The past decade has witnessed great advancements in performance optimization and function extension of on-skin electrodes. With continuous development and great promise for practical applications, on-skin electrodes are playing an increasingly important role in EP monitoring and human-machine interfaces (HMI). In this review, the latest progress in the development of on-skin electrodes and their integrated system is summarized. Desirable features of on-skin electrodes are briefly discussed from the perspective of performances. Then, recent advances in the development of electrode materials, followed by the analysis of strategies and methods to enhance adhesion and breathability of on-skin electrodes are examined. In addition, representative integrated electrode systems and practical applications of on-skin electrodes in healthcare monitoring and HMI are introduced in detail. It is concluded with the discussion of key challenges and opportunities for on-skin electrodes and their integrated systems.

https://onlinelibrary.wiley.com/doi/pdf/10.1002/advs.202001938

Materials, Devices, and Systems of On‐Skin Electrodes for Electrophysiological Monitoring and Human–Machine Interfaces

Bioinspired Multiscale Wet Adhesive Surfaces: Structures and Controlled Adhesion

We demonstrate here that water can be efficiently wet and pumped through superhydrophobic aligned multiwalled nanotube membranes by application of a small positive dc bias. At a critical bias (∼1.7 V), with the membrane acting as anode, there is an abrupt transition from a superhydrophobic to hydrophilic state. Interestingly, this phenomenon is strongly polarity dependent; for a negative bias applied to the membrane, 2 orders of magnitude higher bias is required for the transition. The polarity and voltage-dependent wetting that we report could be used to controllably wick fluids through nanotube membranes and could find various applications.

Polarity-dependent electrochemically controlled transport of water through carbon nanotube membranes.

Drawing inspiration from the adhesion abilities of a leaf beetle found in nature, we have engineered a switchable adhesion device. The device combines two concepts: The surface tension force from a large number of small liquid bridges can be significant (capillarity-based adhesion) and these contacts can be quickly made or broken with electronic control (switchable). The device grabs or releases a substrate in a fraction of a second via a low-voltage pulse that drives electroosmotic flow. Energy consumption is minimal because both the grabbed and released states are stable equilibria that persist with no energy added to the system. Notably, the device maintains the integrity of an array of hundreds to thousands of distinct interfaces during active reconfiguration from droplets to bridges and back, despite the natural tendency of the liquid toward coalescence. We demonstrate the scaling of adhesion strength with the inverse of liquid contact size. This suggests that strengths approaching those of permanent bonding adhesives are possible as feature size is scaled down. In addition, controllability is fast and efficient because the attachment time and required voltage also scale down favorably. The device features compact size, no solid moving parts, and is made of common materials.

https://www.pnas.org/content/pnas/107/8/3377.full.pdf

Capillarity-based switchable adhesion.

Centre-of-mass motions of two coupled spherical-cap droplets are considered. A model with surface tension and inertia that accounts for finite-amplitude deformations is derived in closed form. Total droplet volume A and half-length L of the tube that connects the droplets are the control parameters. The model dynamics reside in the phase-plane. For lens-like droplets λ 1, the symmetric state loses stability (saddle point) and new antisymmetric steady states arise about which limit-cycle oscillations occur. These mirror states - big-droplet up or big-droplet down - are also stable. In addition, there are large finite-amplitude 'looping' oscillations corresponding to limit cycles that enclose both steady states in the phase-plane. All three kinds of oscillations are documented in an experiment that sets the system into motion by 'kicking' one of the droplets with a prescribed pressure-pulse. Model predictions for frequencies are consistent with observations. Small-amplitude predictions are placed in the wider context of constrained Rayleigh vibrations. A model extension to account for the small but non-negligible influence of viscosity is also presented.

Capillary dynamics of coupled spherical-cap droplets

This paper presents a new method to determine the zeta potential of porous substrates in contact with a liquid. Electroosmosis, arising near the solid/liquid boundaries within a fully saturated porous substrate, pumps against the capillary pressure arising from the surface tension of a droplet placed in series with the pump. The method is based on measuring the liquid/gas interface deflection due to the imposed electric potential difference. The distinguishing features of our technique are accuracy, speed, and reliability, accomplished with a straightforward and cost-effective setup. In this particular setup, a bistable configuration of two opposing droplets is used. The energy barrier between the stable states defines the range of capillary resistance and can be tuned by the total droplet volume. The electroosmotic pump is placed between the droplets. The large surface area-to-volume ratio of the porous substrate enables the pumping strength to exceed the capillary resistance even for droplets small enough that their shapes are negligibly influenced by gravity. Using a relatively simple model for the flow within the porous substrate, the zeta potential resulting from the substrate-liquid combination is determined. Extensive measurements of a borosilicate substrate in contact with different aqueous electrolytes are made. The results of the measurements clarify the influence of the ionic strength and pH value on the zeta potential and yield an empirical relationship important to engineering approaches.

Determination of the zeta potential of porous substrates by droplet deflection. I. The influence of ionic strength and pH value of an aqueous electrolyte in contact with a borosilicate surface.

The surfactant concentration distribution on a planar uniform flow with a surface-piercing barrier was measured via the nonlinear optical technique of second-harmonic generation. The measurements were performed for an insoluble surfactant monolayer on the air/water interface. A theoretical model balancing surface elasticity and bulk shear at the interface was developed to predict the concentration profile for any insoluble monolayer. Measured equations of state, relating the surface tension to the surfactant concentration, were used in the model along with velocity data obtained using boundary-fitted digital particle image velocimetry. Theoretical concentration profiles were in agreement with experimental results. Additionally, global predictions from the model for four different insoluble surfactant systems also showed agreement with experimental measurements.

Concentration measurements downstream of an insoluble monolayer front

Dynamics and stability of volume-scavenging drop arrays: Coarsening by capillarity

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